Dirichlet and Liouville integral formulas

Compute the Dirichlet integral

where $V$ is the region bounded by the planes $x+y+z=1$, $x=0$, $y=0$, $z=0$, given the substitutions

Find

Find

Find the volume of the $n$-dimensional simplex

Prove the Dirichlet formula

Prove the Liouville formula

where $f(u)$ is a continuous function.

Find the multiple integral $\displaystyle \int \cdots \int dx_{1} \, dx_{2}\ldots dx_{n}$, taken over the region where all $x_{k}>0$ and $x_{1}+x_{2}+\ldots +x_{n}\leqslant 1$.

Find the multiple integral $\displaystyle \int \cdots \int x_{1} \, dx_{1} \, dx_{2}\ldots dx_{n}$, taken over the region where $x_{k}>0$ and $x_{1}+x_{2}+\ldots +x_{n}\leqslant 1$.

Find the multiple integral

If

then it is easy to prove that $u_{n}(a)=a^{\alpha_{1}+\alpha_{2}+\ldots +\alpha_{n}}v_{n}$, where $v_{n}=u_{n}(1)$. Using this and the equality

prove that

Reduce to a single integral the multiple integral

taken over the region defined by the inequalities

Compute the integral $\iint_{G} x^{m} y^{n} z^{p} d x d y d z$, where $G = \left\{ x^{2}+y^{2}+z^{2} \leqslant 1\right\}$, and $m, n, p$ are nonnegative integers.

Express in terms of values of the $\Gamma$-function the Dirichlet integral

where $G = \left\{ x > 0, y > 0, z > 0, x+y+z < 1\right\} , p > 0, q > 0, r > 0, s > 0$.

Let $S_{n} = \left\{ \sum_{k = 1}^{n} x_{k} \leqslant 1, x_{k} \geqslant 0, k = 1, \ldots , n\right\}$. Derive the Dirichlet formula $\left(x = \left(x_{1} ; \ldots ; x_{n}\right)\right)$

Let $S_{n}$ be the same pyramid as in problem $191$, and let $f(t)$ be a function continuous for $t \geqslant 0$. Derive the Liouville formula

Let $a_{0}, \ldots , a_{n}$ be positive numbers, $b_{0}=a_{1}$, $b_{k}=a_{k+1}-a_{k}$ for $k=1, \ldots , n\left(a_{n+1}=a_{0}\right)$,

(continued).

Define $(u, v) = T(s, t)$ on the strip

by setting $u = s-s t, v = s t$. Show that $T$ is a $1-1$ mapping of the strip onto the positive quadrant $Q$ in $\mathbb {R}^{2}$. Show that $J_{T}(s, t) = s$.

For $x > 0, y > 0$, integrate

over $Q$, use Theorem 10.9 to convert the integral to one over the strip, and derive formula (96) of Chap. 8 in this way. (For this application, Theorem 10.9 has to be extended so as to cover certain improper integrals. Provide this extension).

Let $r_{1}, \ldots , r_{k}$ be nonnegative integers, and prove that

Hint: Use Exercise 12, Theorems 10.9 and 8.20. Note that the special case $r_{1} = \cdots = r_{k} = 0$ shows that the volume of $Q^{k}$ is $1 / k!$.

Let $f: \mathbb {R} \rightarrow \left[0,+\infty \right]$ be $\mathscr {L}^{k} / \mathcal{B}^{*}$-measurable, and let $c_{1}, \ldots , c_{k}$ denote positive real numbers. Writing $\boldsymbol {x}=\left(x_{1}, \ldots , x_{k}\right)$, show that $\boldsymbol {x} \mapsto f\left(x_{1}+\cdots +x_{k}\right)$ on $\mathbb {R}^{k}$ is $\mathscr {L}^{k} / \mathcal{B}^{*}$-measurable and

Let $f$ be as in problem 10.3.7. Let $b_{1}, \ldots , b_{k}$ denote positive real numbers, and write $x_{i}=y_{i}^{1/b_{i}}$ for $i=1, \ldots , k$, obtaining the identity

Show, as in paragraph (4.4.8), that the beta function satisfies $B(a, b) = \Gamma (a) \Gamma (b) / \Gamma (a+b)$.